a singular nonlinear eigenvalue problem: bifurcation in non ... - EPFL

Abstract
This work is concerned with the study of the buckling of a tapered rod. .This physical
phenomenon leads to the following nonlinear eigenvalue problem called problem P:
{A(s)ul(s))' + p sin u(s) = O for al1 s E (O, 1) ,
~(1) = lim A(s)ul(s) = O
1
s-io
1
and A(s)u'(s)~~s < ml
where ~(s) E C([O, 11) is such that A(s) > O for al1 s > O and lim,,,, A(s)/sP = L for
some constants p 2 O and L E (O, CO). We study the set of al1 solutions of this problem
and, in particular, find the points p E IR+ such that bifurcation occurs at (p, 0).
Stuart proved in [18] that there is a number A(A) 2 O such that, for O 5 p 5 A(A),
the only solution of problem P is the trivial solution u = O and it minimizes the energy
in the Hilbert space of al1 admissible configurations denoted by HA. For p :> A(A), there
exists a nontrivial solution of problem P minimizing the energy. Moreover, A(A) > O
for p E [O, 21 whereas A(A) = O for p > 2.
For O 5 p < 3, problem P can be written in the form F(p, u) = O, where F :
I$. x HA -+ HA is such that F(p, U) = u - pGA(u) and GA : HA -+ HA is a completely
continuous operator.
If O 5 p < 2, the operator F is Fréchet differentiable and, using standard tools
like the theorem of Crandall-Rabinowitz concerning bifurcation from simple eigenvalues
and the theorem of Rabinowitz concerning global bifurcation, Stuart was able to prove
that global bifurcation from the trivial solution u r O occurs only at a d.iscrete set of
eigenvalues {p~'
: i E IT), where pl = A(A), pi < pi+l for al1 i E IV, and. limi+oo pi =
CO. Moreover, these eigenvalues are strongly related to the spectrum of the linearization
(in the sense of Fréchet) of GA.
For p = 2, F is not Fréchet differentiable anymore, but only Gâteaux differentiable.
Then, standard results like the ones mentioned above do not apply. However, Stuart
was able to show that there is a number &(A) E [A(A), CO) such that bifurcation from
u = O occurs at every value p E [&(A), CO).
A big part of Our work is devoted to the case p = 2, where it is assumed that
A(A) < A,(A). In this case, there exist eigenvalues {,UT' : i E I C N*) related to
the spectrum of the linearization (but now in the sense of Gâteaux) of GA such that
A(A) = pl, pi < pi+l and A(A) 5 pi < &(A) for al1 i E I. We prove that (local)
bifurcation from the trivial solution occurs at pi for al1 i E I. Moreoier, under an
additional assumption on A(s), we also prove that bifurcation from u rz O does not
occur at p E [O, &(A)) if p # pi for al1 i E I (this is not a standard result since F is not